#72263823

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,11)(3,14,15,20)(4,17,19,9)(6,12,21,31)(7,18,8,13)(10,27,22,26)(16,29,23,30)(24,28,25,32);(1,3,7,9)(2,4,18,14)(5,15,8,17)(6,16,24,26)(10,28,29,12)(11,19,13,20)(21,23,25,27)(22,32,30,31);(1,4,5,19)(2,9,11,17)(3,13,15,18)(6,22,24,30)(7,14,8,20)(10,25,29,21)(12,16,28,26)(23,32,27,31);(1,3)(2,10,11,22)(4,12,19,31)(5,15)(6,23)(7,9)(8,17)(13,30,18,29)(14,28,20,32)(16,21)(24,27)(25,26)

Polycyclic presentation:

2^10|a^2=hij,b^2=hi,c^2=e,d^2=egij,f^2=j,b^a=bhi,c^a=ce,c^b=chij,d^a=defj,d^b=dhi,d^c=dfg,e^d=ei,f^c=fh,f^d=fhj,f^e=fj,g^a=ghi,g^c=ghi,g^d=gi,g^f=gj,h^a=hj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263825, #72263829, #72263833, …
maximal normal subgroup of	#72263825, #72263829, #72263833, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8*F.9*F.10);;
gap> SetPower(coll, 2, F.8*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);